C₁₂₇×#7525 #44571198

order	32512
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₂₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₂₇×#7525

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₂₇	#1404
commutator	#125	C₂×C₄×C₁₂₇
Frattini, cosocle	#491	C₂²×C₁₂₇
prime radical	#562×C₁₂₇	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₂₇ ◁ C₂²×C₁₂₇ ◁ C₂×C₄×C₁₂₇ ◁ C₄²×C₁₂₇ ◁ #491×C₁₂₇ ◁ G

138 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 30 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 159
transitive degree	μ_t(G) = 4064

A minimal faithful permutation representation — 2 generators on 159 points:

(1,2,4,8,5,10,16,26)(3,11,13,24,12,22,29,31)(6,9,20,27,18,28,17,25)(7,15,21,19,23,30,32,14)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159);(2,7)(3,12)(6,18)(8,24)(9,22)(10,23)(11,28)(15,30)(17,20)(21,27)(25,32)(26,31)

Conjugacy classes & character table

2413 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32512, 513);